Axisymmetric finite element analysis of pile loading tests

Computers and Geotechnics 36 (2009) 6–19

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Axisymmetric finite element analysis of pile loading tests I. Said 1, V. De Gennaro *, R. Frank Université Paris-Est, Navier, Ecole des Ponts – CERMES 6-8 Avenue Blaise Pascal, Cité Descartes, Champs-sur-Marne, 77455 Marne-la-Vallée, Cedex 2, France

a r t i c l e

i n f o

Article history: Received 2 December 2007 Received in revised form 18 February 2008 Accepted 20 February 2008 Available online 25 April 2008

Keywords: Interface behaviour Soil–structure interaction Piles Finite elements Installation effects Elastoplasticity

a b s t r a c t In this paper a typical soil–structure interaction problem is considered, the case of a vertical pile installed in sand and submitted to an axial compression loading. Results from two full scale pile tests are analysed and the tests are reproduced by numerical simulations via finite elements method (FEM). The choice of the mechanical parameters for the soil and the sand–pile interface and the modelling approach are first described. A new numerical strategy is outlined to account for pile installation effects due to jacking and driving via FEM. The proposed approach is based on the application of existing empirical correlations available for the quantification of residual radial and shear stresses along the pile shaft as well as residual pressures around the pile base after the installation. This approach is proposed as an alternative to more complex methods based on the numerical modelling of the pile penetration problem. The role of the constitutive modelling of the interface is also discussed. Finally, comparative analyses of pile loading tests using FEM are provided and the comparisons between numerical and experimental results are presented and discussed. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction Nowadays many geotechnical structures such as soil reinforcements and piles are used to avoid failure or excessive deformations of constructions. In this work emphasis is given on the behaviour of deep foundations and, more particularly, of the contact between a granular soil and a pile. During installation and under service loads, piles generate in the surrounding soil stresses and strains, by means of two main mechanisms: the mobilisation of base resistance and lateral friction. The relative importance shared between these two components of pile behaviour depends, among other factors, on the type of applied loads (i.e. traction, compression, lateral, etc.), the pile type and dimension, the type of soil, the installation procedure. In this typical soil–structure interaction problem available analyses of the mechanical behaviour of single piles submitted to axial loads have shown that the soil–pile interface exerts significant influence in defining structural stability conditions. The most fundamental aspects of pile analyses still rely on empirical correlations based on experimental observations from laboratory and full scale in situ testing (e.g. [34]). In both cases, the investigations have been often carried out using instrumented piles, allowing for a direct quantification of the base pressure and the shaft friction. Although proposed empirical correlations allow to roughly quantifying the expected bearing capacities of piles

embedded in various soils, they are not sufficient to assess the associated deformation patterns. For this reason, FEM numerical modelling is often adopted to get a deeper understanding of the soil movement, the pile behaviour and especially the mechanical behaviour of the soil–pile system (et.g. [29,28,4,31,38,14,21,10,39]). However, numerical simulation of test on piles often leads to errors and inaccuracies. This is due essentially to the difficulty of taking account of installation effects and reproduce soil structure interface behaviour. These two problems will be analysed in this paper. The results presented herein are obtained from 2D axisymmetric numerical analyses via finite elements of two full scale tests on piles jacked into sand and subjected to an axial compression load [8,7]. The aim of the calculations was to simulate the mechanical behaviour of the pile considering explicitly: (i) the soils and the interface geotechnical characterization, and (ii) the installation effects in terms of stress changes within the interface layer and the surrounding soils mass. The choice of the mechanical parameters for the soil and the interface, as well as the modelling strategy are highlighted. The available experimental data are used to reproduce the initial state of stress in the soil mass due to jacking. This simulation, although not representative of the overall installation process, is able to encompass some of the aspects related to installation effects. 2. Interface modelling: MEPI model

* Corresponding author. Tel.: +33 1 64 15 35 52; fax: +33 1 64 15 35 62. E-mail address: [email protected] (V. De Gennaro). 1 Present address: ENIT-URIG (Ecole Nationale d’Ingénieurs de Tunis, Unité de Recherche en Ingénierie Géotechnique), Tunisia. 0266-352X/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2008.02.011

The numerical simulation of pile load tests uses the interface model proposed by De Gennaro and Frank [13]. The model, called

I. Said et al. / Computers and Geotechnics 36 (2009) 6–19

MEPI-2D, is based on a Mohr–Coulomb oriented failure criterion including the principal characteristics of sand–structure interface such as mobilised friction hardening–softening, phase transformation state and ultimate state. The formulation of the model is two-dimensional; it associates normal and tangential relative displacements components un and ut of the incremental relative displacements vector du = (un,ut)T to normal and shear stress components rn and s of the incremental stress vector dr = (drn, ds)T. The elastic behaviour of the interface is given by:   Kn 0 ð1Þ dr ¼ Ke due ; Ke ¼ 0 Kt where Kn and Kt are the normal and tangential stiffness of the interface. The yield surface F (assuming compression positive and neglecting cohesion) depends on the hardening function lðupt Þ. It can be written as: F ¼ s  lðupt Þrn ¼ 0

ð2Þ

7

Parameter B defines the rate of dilatancy stabilization at the interface. When B increases dilatancy stabilization at the zero value (i.e. ultimate state) is more rapid. Note that if D = 1 in Eq. (4) the single flow rule of the original Cam-clay model is obtained. The following phenomena observed experimentally on sand–structure interfaces can be considered using MEPI-2D model: (1) the presence of an initial compaction (dun > 0) in a test with constant normal stress or a reduction of the stress rn normal to the interface layer in a test with imposed constant volume or constant normal stiffness; (2) the presence of the transformation phase from compaction to dilatancy ( dun < 0) which corresponds to the increase of the normal stress in a test with imposed constant volume or constant normal stiffness; (3) the stabilisation of the normal relative displacement un, or the normal stress rn, on a asymptotic value for large relative tangential displacements at the interface (dun = 0 or drn = 0), which corresponds to the ultimate state, where constant volume conditions are assumed at failure.

where up lðupt Þ ¼ lo þ ðlf  lo Þ   t rni A p t þ upt

ð3Þ

o

The hardening function lðupt Þ gives the evolution of the mobilised friction coefficient l during loading; it is assumed of hyperbolic type. The tangential plastic displacement upt is the hardening parameter, lf is the friction coefficient at failure, lo defines the elastic region, A is a parameter controlling the rate of the deviatoric hardening, t is the thickness of the interface layer (usually assumed equal to 10–15 times the average grain diameter, D50), rni is the initial normal stress acting on the interface, and po is a reference pressure (1 kPa). Note that only hardening is accounted herein, that is: lf the friction coefficient at failure is equal to lr the friction coefficient at ultimate state; the basic version of the model includes also softening behaviour [13]. The plastic flow rule of MEPI-2D model is an extended form of the stress–dilatancy relation d = M  g given by Roscoe et al. [36] in the formulation of the original Cam-clay model. The flow rule has p u_ been defined using the dilatancy ratio d ¼ u_ np and considering that t the stress ratio lc at phase transformation (transition from compaction to dilatancy) in sand interfaces does not coincide with the value of lr at ultimate state, although for both situations condition d = 0 is fulfilled. Lings and Dietz [25] and Dietz and Lings [17] suggest that interface apparatuses compliances could be at the origin of this behaviour, even if the same behaviour has been observed in sands tested using a triaxial cell [24], where apparatuses compliances are less severe. In the circumstance that lc and lr are distinct characteristic for the same value of dilatancy (d = 0) the hypothesis of a single flow rule (i.e. dilatancy d is a unique function of l) is no longer valid and the dependency of lc on a state parameter of the sand has to be considered (e.g. [11,24]). The extended form of the stress–dilatancy relations proposed in MEPI-2D model accounts for this situation admitting that dilatancy depends on a state parameter, which is represented by the relative plastic tangential displacement upt (e.g. [12,15]). The following stress–dilatancy relationship is considered: d¼

u_ pn ¼ ðlco  lÞDðupt Þ u_ pt

ð4Þ

where lco is the stress ratio at phase transformation state, l is the mobilised friction coefficient and D ¼ Dðupt Þ is given by:     1 B rni upt ð5Þ Dðupt Þ ¼ cosh t po

As formulated MEPI-2D model requires eight parameters: kn, kt, lo, lf, lco, A, B, and po; being rni, and t given by the initial conditions. The validation of the proposed approach for interface modelling has been performed using tests at constant normal stress, tests at constant volume and tests at constant stiffness [12,13]. 3. Numerical modelling of full scale model pile test The first pile test results analysed are those reported by Chow [8] using the Imperial College Pile (ICP). The steel ICP used during the test has a diameter D = 0.102 m, and a length Lp = 7.4 m (Fig. 1a). The pile is close ended with a 60° conical shape. Due to the reduced diameter ICP can be considered as a model pile rather than a real pile. The instrumentation is concentrated in four clusters (Leading, Following, Trailing, Lagging; see Fig. 1a) about 1 meter spaced along the shaft, the first cluster being 0.2 m distant from the pile tip. Each cluster contains: (i) an axial load cell measuring the axial load transmitted through the pile and the average shear stress between two clusters (fs), and (ii) a surface stress transducer, allowing the measurement of the total radial stress (rr) and the local shear stress (srz). Further details on ICP instrumentation can be found in Bond [3] and Chow [8]. The pile was installed in a rather homogeneous formation of dense Flandrain sand (Dr ffi 75%). The Flandrian sand horizon is interrupted by an organic layer 0.6 m thick at 7.6 m depth. The effect of this soft layer has been neglected in this study. The water table was at 4 m depth (Fig. 1b). 3.1. Finite element mesh and boundary conditions Numerical simulations were performed using the software CESAR-LCPC [23]. The geometry of the problem, the geotechnical data and the finite element mesh used for the simulations of the pile test are shown in Fig. 1b. It is defined for axisymmetric conditions (Z being the axis of symmetry, coinciding with the pile axis). The limits of the domain investigated are fixed at 30 m in the vertical direction (i.e. more than three times the pile length, Lp) and three times the pile length in the radial (horizontal) direction. It consists of 1850 axisymmetric isoparametric eight-node solid elements, including 100 pile elements and 50 interface elements. The interface layer is 3 mm in width (12 times D50 of the sand) and its behaviour is simulated by means of eight-node thin-layer elements [16]. MEPI-2D constitutive law was assigned to the thin-layer elements during pile loading. The aspect ratio L/t of the interface elements (L/t = 48) was chosen in order to reproduce

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Fig. 1. Model pile test in Dunkirk: (a) instrumented IC pile (after [8]) (b) geometry and geotechnical data and (c) mesh and boundary conditions.

correctly the interface behaviour and avoid numerical ill-conditioning [33,22,12]. Horizontal displacements are set to zero on the external vertical boundary, whereas on the bottom limit only the vertical displacements are set to zero. 3.2. Constitutive modelling of sand and sand–steel interface The Flandrian sand from Dunkirk is supposed elastic perfectly plastic, obeying the Mohr–Coulomb failure criterion. The mechanical parameters were obtained from data by Kuwano [27], quoted in Chow [8], who performed triaxial tests and interface direct shear tests on this sand. The elastic parameters are Es = 200 MPa and ms = 0.3. Values of Es were found considering in situ measurements by means of CPT correlations [8]. Besides an initial increase in the first two meters, these values showed an increase with depth of about 0.9 MPa/m. For this reason the values of Es were considered almost constant all over the pile length (7 m). The internal friction angle is /0 = 37° and the dilatancy angle is w0 = 10°. Linear elasticity was considered for the ICP, with Ep = 195 GPa and mp = 0.28 (steel pile). The interface behaviour between Dunkirk sand and ICP is simulated using MEPI-2D model. The parameters of the model were de-

rived from the results of the interface tests performed by Kuwano [27]. Tests were conducted on sand–steel interface in a 60 mm  60 mm modified direct shear box under constant normal stress level ranging between 40 and 300 kPa and with a shear rate of 0.5 mm/min. A dense sample of Flandrian sand and a steel plate of roughness R = 10 lm (maximum notch depth) were used. Fig. 2 shows typical experimental results of constant normal stress interface test (rn = 150 kPa) in terms of interface shear stress vs relative tangential displacement (Fig. 2a) and normal relative displacement vs relative tangential displacement (Fig. 2b). The graphs show a peak shear stress reached after a horizontal displacement of less than 1 mm. The evolution of the normal displacement shows a small initial compaction followed by dilatancy and constant volume conditions corresponding to large tangential displacements (ultimate state). The set of MEPI-2D model parameters was determined from the interface test results plotted in Fig. 2. Numerical simulations of the interface test are also plotted in the same figure (bold lines). The parameters are summarized in Table 1. Due to the lack of experimental data the normal stiffness of the interface Kn was assumed equal to 2Kt. The comparison between experimental results and simulations is satisfactory with regards to the evolution of the shear stress s and the normal displacement un versus the

Fig. 2. Comparison of model prediction and experimental results: interface test at constant normal stress (rn = 150 kPa) carried out on a modified direct shear box (Dense Dunkirk sand, experimental data after [27]).

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et al. [26], that gives the final value of the radial stress along the pile shaft after the equalisation period, it reads:

Table 1 Constitutive parameters for steel-Dunkirk sand interface (rn = 150 kPa) Kn (kPa m1)

Kt (kPa m1)

lo

lf

lco

A

B

t (mm)

501612

250806

0.017

0.51

0.25

0.00008

0.065

3

tangential displacement ut at the interface (Fig. 2). Note that the softening phase observed experimentally is not captured by the numerical simulation since the implemented version of the model does not include strain-softening. As already mentioned MEPI-2D model was associated to the eight-node thin-layer elements used to simulate the interface layer. This was done following the approach proposed by Frank et al. [19] and further developed by Sharma and Desai [37]. In this approach the two stress components s and rn acting at the interface are calculated in each integration point of the thin-layer element as a projection on the directions perpendicular and parallel to the interface layer of the general stress tensor acting on that point. In other words stresses are ‘‘oriented” in the direction of the interface and the interface constitutive law links the two components of stress rn and s, perpendicular and parallel to the interface layer. During the initialisation of the state of stress in the soil (geostatic stresses and stresses induced by installation effects), the interface layer was simulated assuming Mohr–Coulomb oriented failure criterion and non-associated plasticity. MEPI-2D model was used to simulate the interface behaviour during pile loading test. 3.3. Calculation phases In order to simulate the overall behaviour of the pile prior and during the loading test three phases were identified and reproduced numerically using FEM. These phases are summarised in Fig. 3, they correspond to the geostatic initialisation of the state of stress within the soils mass (Fig. 3a), the simulation of the installation effects (Fig. 3b) and, finally, the simulation of pile loading test (Fig. 3c). 3.3.1. Geostatic initialisation To reproduce the initial conditions in terms of effective stresses in the soil mass before loading (step 1, Fig. 3a), sand weights c above and below the water table (4 m depth) were activated (geostatic initialisation) using the geotechnical data of Fig. 1. The initial horizontal stress was taken into account numerically considering 0 the K0 condition given by K 0 ¼ ð1  sin u0 ÞOCRsin u [30]. As indicated by Chow [8], in Dunkirk site Flandrian sand was normally consolidated (OCR = 1). The state of stress within the soil mass at the end of this phase is designated as r1o;s , while r1o;i and r1o;p represent the state of stress within the interface and below the pile tip, respectively. 3.3.2. Installation effects On the site, pile installation was performed ‘‘step by step”, using a hydraulic jack, imposing a constant pile head displacement rate of 600 mm/min. At the end of the installation, an equalisation period of about 15 h was considered, based on the time necessary to stabilise the values of the measured radial effective stress. Three important phenomena were observed during jacking and were considered numerically in this numerical step: (i) the increase of the radial effective stress along the pile shaft, (ii) the mobilisation of a negative residual friction along the pile shaft and (iii) the residual pressure around the pile base. The overall increase of the radial effective stress r0rc along the pile shaft at the end of the equalisation period was imposed considering the empirical formula proposed by Chow [8] and Jardine

r0rc ¼ 0:016qc

 0 0:13  0:38 rvo h pa R

ð6Þ

where h is the vertical distance from the pile tip, R is the pile radius, r0vo the effective vertical stress, r0rc the radial effective stress at the end of installation and qc is the net cone resistance from CPT and pa is a reference pressure equal to 1 kPa. The applicability of Eq. (6) for jacked and driven piles has been verified on the basis of a quite large number of high quality pile tests (e.g. [34,26]) and could be considered well representative of the distribution that one might get around a pile installed in sand. The radial effective stress r0rc was imposed numerically as an external load on the interface layer considering a linearization of stresses in various horizontal soil horizons (Fig. 3b). At the end of this calculation the radial effective stress values after pile installation matched the measured ones, allowing the reproduction of the final effect of jacking on the radial confinement of the pile. The distribution of radial stress against depth imposed during the numerical simulations is compared satisfactorily to the experimental values in Fig. 4a. The distribution of the negative residual friction imposed along the pile shaft is shown in Fig. 4b. The shape of the curve can be approximated by a bilinear fitting as suggested by Alawneh and Malkawi [1], with the point with zero residual friction close to the pile tip. Note that this fitting is only valid for long or flexible piles. The values of residual friction sres were imposed numerically along the pile shaft (Fig. 3b), simultaneously with the application of the radial stress after installation given in (6). This initialisation led to the final state of stress r2o;i (Fig. 3b). Finally, the residual pressure developed below the pile tip during installation was considered (r2o;p , Fig. 3b). The value of qp-res results from the equilibrium of the pile at the end of jacking considering that the action of the residual base pressures (upward) balances out the overall negative residual shaft friction (downward) and the pile weight. This change in base load after installation was considered during the numerical analyses imposing at the pile base the pressure qp-res ffi 6300 kPa measured at the end of the equalisation period [8]. Alternatively, in the absence of direct measurements, the value qp-res can be determined adopting available empirical formulations (e.g. [5,1]). It is to be noted that during step 2, the pile elements are not yet activated. 3.3.3. Loading test Step 3 consisted in the simulation of the static loading test. Compression loads were applied following a variant of the LCPC procedure given by Bustamante [6]. On the site loads were applied by successive increments of 5–10% of the expected maximum load, and maintained for 10–20 min. Failure was reached when the rate of pile movement increased significantly with time. The failure axial load was 309 kN, which corresponds to a pile head displacement of 4.2 mm [8]. Numerically, loading is applied by activating the pile elements (weight and stiffness) and applying displacement increments at the pile head (Step 3, Fig. 3c). During this phase the state of stress is modified in the soil mass (r3o;s ), the interface (r3o;i ) and the pile tip (r3o;p ). 3.4. Discussion of results The total head displacement during compression loading was achieved in 80 increments, (during each increment 1.25% of the final vertical displacement was applied). With regard to the mobilisation of the shear stress along the pile shaft, the results of

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Fig. 3. Calculation phases for FEM modelling of jacked/driven pile tests: (a) geostatic initialisation, (b) installation effects and (c) loading test (all stresses are effective stresses).

calculations are compared in Fig. 5 with the experimental data. Measurements refer to the shear stress s measured locally, in each of the cluster located on the ICP shaft (Fig. 5a) and to the average shear stress at the pile shaft sav (Fig. 5b). We may notice that the experimental curves start from a residual negative friction due to the relaxation of the soil–pile system after jacking. As mentioned earlier, shear stress changes on the pile shaft due to jacking were considered during step 2 in FEM calculations (Fig. 3). The shape of the mobilised shear stress curves appear in good agreement with the measured ones. The final plateau of the predicted shear stress curves tend to be overestimated for the Following cluster measurement, and slightly underestimated for the Lagging cluster measurements (Fig. 1a). Nonetheless, the average shear stress along the pile shaft compares well with the experimental value. The average shear stress found at failure is sav = 89 kPa against an experimental value of about 92 kPa (Fig. 5b).

The predicted initial stiffness of the interface, given by the initial slope of the shear curves in Fig. 5a, is also in good agreement with the measurements. The initial stiffness depends on the contractancy and/or dilatancy of the interface layer and, furthermore, on the surrounding soil stiffness. It seems then that the constitutive parameters of the interface and the surrounding soil are realistic. It is observed that the interface tests revealed strong dilatant behaviour [8], in agreement with the high density of the sand deposit (Dr = 75%). These data were further corroborated during loading test, where an average increase of the radial stress Dr0r ¼ 60 kPa was measured. As shown in Fig. 6a, the dilatant behaviour of the interface is well captured by the numerical simulations. Clearly, the increase of the radial stress, normal to the interface layer, illustrated in Fig. 6a, has a major role in the mobilised shear stress curves presented in Fig. 5a. The prediction of the axial capacity of the pile during the test is shown in Fig. 6b. Both the pile head load (Qt) vs the pile head dis-

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Fig. 4. (a) Numerical conditions imposed along the pile shaft at the end of the installation: (a) radial effective stress (b) negative residual friction (data after [8]).

Fig. 5. Mobilisation of the shear stress during loading: (a) local values and (b) average value along the pile shaft.

Fig. 6. Numerical predictions vs. experimental results ICP test: (a) radial stress in the four clusters along the pile shaft during loading (experimental data not available) and (b) load displacement responses during loading (without densified zone below pile base).

placement and the end-bearing load (Qp) are compared with the results from numerical simulation using FEM. Base reaction develops starting from the value of Qp imposed at the end of installation (about 51 kN). However, there appears to be a discrepancy be-

tween the measured base resistance at failure (92 kN) and the predicted one (about 71 kN), corresponding to a pile head displacement of about 4 mm. It’s quite likely that the discrepancy between predicted and measured base resistance at failure is the

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consequence of neglecting a basic phenomenon associated to the installation by driving/jacking: the densification of the sand around the pile base. Indeed, this induces an increase in the soil stiffness which is not taken into account by the numerical simulations of the pile test. The same phenomenon takes place also along the pile shaft, but it is partially encompassed numerically by means of the application of the radial stress given in (6). In an attempt to account for the increase of the base stiffness in a simplified way, the Young’s modulus of the zone below the pile base has been modified. The extension of this zone, almost entirely compressed, has been early identified by Robinsky and Morrison [35] and more recently by Allersma and Broere [2], White and Bolton [40], Dijkstra et al. [18]. It appears that during installation the movement of the grains is set on a zone that extend below the pile tip up to a distance of 2 to 3.5 pile diameter D and in the side direction from 3D to 4D from the pile axis (Fig. 7a). It was then decided to consider a rectangular zone 3.5D height and 4D large and increase the Young’s modulus of this zone before starting the simulation of the loading test (step 3) in order to account for soil densification due to installation. To quantify the amount of increase of Young’s modulus, the variation of E with density and consolidation stress (rc) established by De Gennaro [12] on Fontainebleau sand during unloading–reloading phases (Eur) in triaxial compression tests were considered (Fig. 7b). The following relation was assumed between Young’s modulus and consolidation stress: Ed ¼ mr0:55 c

ð7Þ

In the absence of similar data for Dunkirk sand, Fontainebleau curves were considered valid. The maximum densification is supposed to be reached for Dr = 100%. In this case, starting from the results of Fig. 7b, a linear interpolation between the state parameter m associated to the current density and the corresponding relative density Dr = 100% gave a value of m ffi 20. The modified Young’s modulus was determined using (7) and considering the value of the radial stress measured in situ close to the pile base after installation ðrc ¼ r0rc ¼ 215 kPaÞ. The modified value was Eur = 420 MPa; compared to the initial value Eo ffi 200 MPa an increase of about 110% was then considered. As shown in Fig. 8 and Table 3 the change in Eur value has a clear effect on the numerical prediction of the pile base load Qp and the application of the new elastic parameters clearly improve the numerical predictions at failure.

Fig. 8. Numerical predictions vs. experimental results ICP test: load displacement responses during pile loading with densified zone below the pile base.

Figs. 9 and 10 show the distribution of the vertical displacements and the evolution of the plastic deformation around the pile for various loading steps. A detailed observation of the zone around the pile shows that numerical calculations predict a bulb formed around the inclusion characterized by vertical displacements directed downwards. The lateral extension of this bulb decreases gradually around the inclusion during pile loading. At the end of the loading phase, the bulb is almost completely localised around the pile, in the interface layer, and slightly below the pile base, and vertical displacements become negligible elsewhere. Note that for a better estimation of the shape and the extension of this zone below the pile base a large deformation analysis would have been more suitable. The evolution of the plastic zones (Fig. 10) indicates that the yielding of the interface and the surrounding sand increases for increasing values of the pile settlement (the percentage of the maximum plastic deformations reached during the loading is indicated for each loading step).

Fig. 7. (a) Volumetric strain (compression is negative) during pile penetration in a medium-dense granular material (after [18]) and (b) evolution of Young’s modulus of Fontainebleau sand according to consolidation stress and density [12].

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Fig. 9. Numerical simulation of the evolution of vertical displacements around the ICP during loading.

Fig. 10. Evolution of the plastic deformations around the ICP during loading.

3.5. Influence of the interface model To highlight the influence of the interface model on the results of simulations of Dunkirk pile test, two series of calculations were performed. In the first series the interface behaviour is neglected (i.e. perfect adherence between the pile and the sand is assumed). In the second series perfect plasticity was considered, admitting Mohr–Coulomb failure criterion and non-associated plastic flow ‘‘oriented” in the direction of the interface [19]. The parameters

of this model, known as ‘‘oriented criterion” (OC) are given in Table 2. The comparison of the calculations is presented in Fig. 11. As expected, neglecting the relative displacement between the pile and the sand (perfect adherence) the pile settlement at failure is underestimated (about 1 mm assuming perfect adherence, against a measured settlement of 4.3 mm). If the interface behaviour is considered using the OC model, the experimental load–settlement curve is better reproduced, although the values of settlement remain on the whole underestimated.

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Table 2 Constitutive parameters used for the numerical study of ICP test Parameter Sand (Mohr–Coulomb) Young‘s modulus Poisson’s ratio Cohesion Friction angle Dilatancy angle

Es = 200 Mpa ms = 0.3 cs = 0.1 kPa /0 = 37° w = 10°

Young’s modulus Poisson’s ratio

Ep = 195 Gpa mp = 0.28

Pile (linear elastic)

Interface (oriented criterion) (step 1 and 2: stress initialisation) Ei 200 MPa 0.3 mi 0.1 kPa ci d0i 27° 10° wi Interface (MEPI-2D) (step 3: pile loading) E* m* lf = lr lco lo A B t

1500 kPa 0 0.51 0.25 0.017 0.00008 0.065 3 mm

tained with the OC model are the consequence of the absence of stabilization of the shear stress (Fig. 12). This increase in the shear stress is caused by the non stabilization of dilatancy in the interface layer (w = cste). Indeed, the constant dilatancy predicted by the non-associated Mohr–Coulomb model, constrained by the stiffness of the adjacent sand generates a continuous increase in the radial stress (Fig. 13b) and consequently in the shear stress. This situation is not met in the case of the numerical simulations using MEPI-2D model for the interface because the model considers a stabilization of dilatancy at large tangential displacements (i.e. D ? 0 when ut ? 1, Eq. (5)). The evolution of the average shear stress at the pile shaft (Fig. 13a) confirms the strong dilatant behaviour given by the oriented criterion (CO) and shows that simulations with MEPI-2D model reproduce more satisfactorily the interface behaviour. A synthesis of the various load–settlement curves obtained varying the behaviour of the interface is shown in Fig. 13b. It is noted that the base resistance is the same for all the simulations. The load–settlement curve is well reproduced using MEPI-2D model for the interface and generally over-estimated adopting the oriented criterion ‘‘OC”. Also, the shape of the load–settlement curve shows that the failure condition represented by a final plateau is not fulfilled using OC model. Once again, the effect of a constant constrained dilatancy is at the origin of this behaviour.

4. Numerical modelling of full scale loading pile test Table 3 Comparison of the numerical and experimental loads at failure

Experiment Numerical modelling

Friction load Qs (kN)

Pile base load Qp (kN)

Total load Qt (kN)

217 211

92 89

309 300

The second set of numerical analyses concern a driven box pile. Loading test is described by Bustamante and Gianeselli [7]. A less detailed geotechnical characterization of the site is available for this example. This will allow to checking and better appreciate the extent of previous assumptions, in particular concerning the simulation of installation effects. 4.1. Pile geometry and site characterisation The pile loading test analysed herein was conducted on a driven pile installed in Dunkirk [7]. The pile length is Lp = 7.76 m and its geometry is hexagonal (Table 4, Fig. 14). In order to perform axisymmetric modelling in CESAR-LCPC, the hexagonal pile has been replaced by a cylindrical one, using the equivalence of the lateral surface to determine the equivalent pile diameter. The lateral surface is then expressed as follows: Slat ¼ P  Lp ¼ 2pReq Lp

ð8Þ

where Slat is the lateral surface of the box pile, P is the perimeter of the box pile, Lp is the pile length and Req is the equivalent radius used for the circular pile. Req is found equal to 26 cm. Furthermore, the equivalent Young’s modulus of the pile is determined in order to have the same stiffness as the real pile: Eeq ¼

Fig. 11. Comparison between experimental data and numerical simulations of the load–settlement curve of ICP at Dunkirk, effect of the interface modelling with oriented criterion (OC).

A comparison of local shear and normal stresses at the interface using OC model and MEPI-2D model allows a better understanding of this underestimation (Fig. 12). The mobilization of the local shear stress in the 4 ICP clusters shows that the higher values ob-

ðEAÞpile pR2eq

ð9Þ

where A is the average section of the pile, E is Young’s modulus of the real steel pile (E = 195 GPa). We obtain Eeq = 16 GPa. Thus the circular pile analysed by finite elements is circular, of diameter D = 0.52 m, length Lp = 7.76 m and Young’s modulus E = 16 GPa. The pile was instrumented using the LCPC extensometers and delimiting five measuring sections. Note that, for this test, residual stresses after pile driving along the shaft and below the pile tip are not measured. However, their values have been determined using available correlations and applied numerically. This will be shown in the section devoted to the numerical analysis and discussion of results.

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I. Said et al. / Computers and Geotechnics 36 (2009) 6–19

500

200

Normal stress σn (kPa)

160

Shear stress τ (kPa)

450

MEPI 2D OC

120

80

40

MEPI 2D OC

400 350 300 250 200 150 100 50

0 0

2

6

4

0

8

0

2

Pile head displacement (mm)

-40

4

6

8

Pile head displacement (mm)

Fig. 12. Comparison between simulations using the oriented criterion (OC) and MEPI-2D for interface modelling: (a) local shear stress at clusters and (b) normal stress at clusters.

500

180

450

OC Experiment MEPI 2D

150

350

LOAD (kN)

120

τav (kPa)

QQ (OC) t (OC) t (CO)

400

90 60 30

300

Qt (MEPI 2D)

250

Qt (Exp)

200 150

Qp(Exp)(rupture)

100

Qp(Num)

50

0 0

2

4

6

8

0 0

-30

Pile head displacement, wto (mm)

2

4

6

8

Pile head displacement, wto (mm)

Fig. 13. Comparisons between experimental results and numerical simulations using oriented criterion (OC) and MEPI 2D for the interface: (a) average shear stress against pile settlement and (b) load–settlement curve.

Table 4 Geometrical characteristics of the LCPC box pile B (mm)

H (mm)

E (mm)

S (mm)

Perimeter (cm)

Weight (kg/ m)

Section steel (cm2)

533

385

12.3

342

164

139

177

4.2. Finite element mesh and boundary conditions The finite element mesh used for the simulations of the pile test is shown in Fig. 15. The mesh is similar to that used for the ICP analysis. It consists of 2729 axisymmetric isoparametric eightnode solid elements, including 128 pile elements and 64 interface elements. The interface layer is 3 mm in width and its behaviour is simulated by means of eight-node thin-layer elements. The aspect ratio L/t of the interface elements is fixed at 40. 4.3. Constitutive modelling of sand and sand–steel interface Sand from Dunkirk was supposed elastic perfectly plastic, obeying the Mohr–Coulomb failure criterion. The mechanical

Fig. 14. Geometry of the LCPC box pile [7].

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I. Said et al. / Computers and Geotechnics 36 (2009) 6–19

Z

Pile Interface

Layer 1

w to

couche1 Layer 2 couche2 Layer 3 couche3

L =7, 76 m p

Layer 4 Layer 5

u=0

u=0

v=0

R

Fig. 15. LCPC pile test in Dunkirk: (a) geometry and (b) mesh and boundary conditions.

parameters were obtained from pressuremeter tests results. Young’s modulus is determined from the pressuremeter modulus EM. The analyses of Frank [20], which compares the Ménard pressuremeter and the self-boring pressuremeter (PAF) for the estimation of the pressuremeter modulus EM were considered. It is suggested that:  4:1 for sands Gpo ð10Þ ¼ GM 11:3 for clays where Gpo is the secant shear modulus with PAF corresponding to a strain of 0.02% (almost the initial tangent modulus); GM is the shear modulus with Ménard pressuremeter. The self-boring pressuremeter (PAF) is used to avoid the drawbacks of the Ménard pressuremeter installation which remoulds the soil. Since with PAF, the mechanical characteristics of the soil remain almost intact, it is admitted that the initial modulus of PAF is almost equal to Young’s modulus of the soil (small deformations). Consequently, the following relations between Young’s modulus and the pressuremeter modulus can be written:  4:1 for sands E ð11Þ ¼ EM 11:3 for clays The friction angle of sand is related to the limit pressure of the Ménard pressuremeter as follows [32]: p  p0 u0 ¼ 25 þ 5:77 log LM b

ð12Þ

where pLM is the limit pressure measured with the pressuremeter, p0 is the horizontal total stress before the test (b = 1.8 for wet homogenous soils, 3.5 for dry heterogeneous soils and 2.5 for other soils). The friction angle was also calculated using the empirical formula proposed by Combarieu [9] for sands: 2 3 ln qpl 9 41 0 0 sin u ¼ þ ð13Þ 5 8 8 ln pl þ 3 3 2 q0

where q0 is the initial vertical stress. An average friction angle value obtained from (12) and (13) was then considered in each layer. Young’s modulus and the friction angle values are given in Table 5. Linear elasticity was considered for the LCPC pile (Table 5). The methodology outlined in Fig. 3 was adopted to reproduce the initial conditions prior to pile loading. During the initialisation of the state of stress in the soil (geostatic stresses and installation effect), the interface layer was simulated assuming Mohr–Coulomb oriented failure criterion and non-associated plasticity, like in the case of the ICP test. During the simulation of the loading test, non-associTable 5 Constitutive parameters used for the numerical study of LCPC pile test Parameter

Layer 1

Layer 2

Layer 3

Layer 4

Sand (Mohr–Coulomb) Young’s modulus Es (MPa) Poisson’s ratio m Cohesion: c (kPa) Friction angle: /0 (°) Dilatancy angle w (°)

29 0.3 0.1 35.8 5.8

37 0.3 0.1 38.7 8.7

53 0.3 0.1 41.2 11.2

62 0.3 0.1 40.5 10.5

Pile (linear elasic) Young’s modulus Ep (GPa) Poisson’s ratio mp Interface (Mohr–Coulomb orienté) Ei(MPa) mi ci (kPa) d0i ð Þ wi (°) Interface (MEPI 2D) E* (kPa) m* lf lco lo A B t (mm)

Layer 5

94 0.3 0.1 41 11 Value 16 0.28 200 0.3 0.1 27 10 1500 0 0.51 0.25 0.017 0.00008 0.065 3

I. Said et al. / Computers and Geotechnics 36 (2009) 6–19

ated elastoplastic behaviour was considered at the interface, based on the interface model MEPI-2D. Since no interface test results were available, the same interface constitutive parameter as for the ICP load test were considered, assuming the same density for the Dunkirk sand and the same rugosity for the LCPC pile. 4.4. Numerical analysis and discussion of results 4.4.1. Initial conditions and installation effects Initial conditions in terms of geostatic initialisation and pile installation effects are based on the approach proposed in Fig. 3, as for the previous simulations of the ICP test. The sand weight and horizontal initial stress (Ko condition) are first accounted numerically (Fig. 3a). Pile installation effects are then imposed numerically following Fig. 3b. The residual radial stress was determined along the pile shaft using Eq. (6). In the absence of direct measurements the residual base pressure qp-res was calculated as an average value between the two empirical formulations proposed by Briaud and Tucker [5] and Alawneh and Malkawi [1], Briaud and Tucker [5] formula writes:

qpres ¼ 533:4Lp b sffiffiffiffiffiffiffiffi KsP ; K s ¼ 188:9ðN side Þ0:27 b¼ AEp

17

ð14Þ

where Ks is the initial slope of the friction curve during the loading (kPa/cm), P is the pile perimeter (cm), Ep is the pile elastic modulus (kPa), A is the pile section (m2), Lp is the pile length (m), Nside is the average value of SPT sand blow count around the pile. The second formula [1] considers the pile flexibility: qp-res ¼ 13158g0:724     L Ap G g¼ A D Ep

ð15Þ ð16Þ

where g is the flexibility factor, D is the pile diameter, G is the sand shear modulus, Ap is the cross-section of the pile. The two methods give an average residual base resistance qp-res = 660 kPa. This value was applied at the pile base in the second step (Fig. 3b). Residual shear stress sres after driving is calculated considering the equilibrium between the residual base force, the residual friction and the pile weight W.

Fig. 16. LCPC pile test in Dunkirk, mobilisation of the shear stress during loading: (a) local values and (b) average shaft friction.

Fig. 17. LCPC pile test in Dunkirk: (a) radial stress in the five sections along the pile shaft during loading and (b) load–settlement curves: numerical predictions vs. experimental results (after [7]).

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sres ¼

I. Said et al. / Computers and Geotechnics 36 (2009) 6–19

Aqp-res  W ¼ 10 kPa Slat

ð17Þ

The average residual shear stress obtained was applied numerically as a uniform stress on the interface limits (step 2, Fig. 3b).

Further developments would be necessary at more fundamental level to be able to reproduce the overall installation process. This was out of the scope of this work. References

4.4.2. Loading test Moving from the initial conditions obtained during the previous numerical modelling (step 2), compression loading was simulated in the third step applying a total displacement wto = 50 mm to the pile head (Fig. 3c). The total head displacement was reached in 100 increments, (during each increment 0.5 mm was applied). Note that, as for ICP test, a modified Young’s modulus E ffi 400 MPa in the rectangular sand zone around the pile base has been considered (Fig. 7a), assuming that the sand below the LCPC pile base reaches the same stiffness at maximum density like for ICP test. The results of the calculations of the mobilisation of the shear stress along the pile shaft are compared with the experimental data in Fig. 16. Simulations refer to the shear stress measured locally, in each of the sections located on the pile shaft (Fig. 16a) and to the average shaft friction (Fig. 16b). Comparisons between experimental data and calculations show that numerical curves starts from non-zero residual values. This is not the case for the test, as measurements were set to zero at the beginning of loading test, withdrawing installation effects in terms of residual values. The evolution of experimental shaft friction is well reproduced by the numerical calculations. Like for the ICP test, the increase of the radial stress normal to the interface layer, illustrated in Fig. 17a, has a major role in the mobilised shear stress presented in Fig. 18a. The evolution of the total base resistance Qp and of the total load Qt are compared to the experimental results in Fig. 17b. Although values at failure (large pile head displacements) are well reproduced, the initial slopes of both numerical curves are slightly underestimated. For the base load numerical curve it’s quite likely that the increase in the overall stiffness during loading, related to the sand densification around the pile base after driving is at the origin of this scatter. Concerning the initial slope of the total load numerical curve Qt(Num), the underestimation seems to be related to the scatter observed on the base load numerical prediction. In fact the local behaviour in terms of mobilized shaft friction (Fig. 16) agrees well with in situ measurements. 5. Conclusion In this paper the results of the numerical analysis of load tests on instrumented jacked/driven in situ piles installed in sand are presented. A first attempt to account for installation effects during numerical simulation via finite elements has been shown. Empirical correlations based on field data have been used to reproduce the change in radial stress around the pile, consecutive to jacking/driving. Similar correlations are employed to reproduce the installation effects in terms of residual shaft friction and base resistance. The importance of the constitutive modelling of the interface layer has been clearly illustrated, mostly in relation with the modelling of the volumetric behaviour at the interface, in terms of dilatancy and/or contractancy, using a deviatoric hardening model. Friction at the interface layer is well reproduced numerically. Base resistance results are improved considering the change of material mechanical properties due to pile installation (i.e. soil densification). Although this approach remains quite crude and simple, its efficiency looks satisfactorily from a practical point of view. Load–settlement curves are well reproduced by the numerical modelling and the results seem promising in view of the numerical modelling of other tests on full scale in situ piles.

[1] Alawneh AS, Malkawi AIH. Estimation of post-driving residual stresses along driven piles in sand. Geotech Test J 2000;23(3):313–26. [2] Allersma HGB, Broere W. Optical analysis of stress around a penetrating probe in granular material. In: Philips R, Guo P, Popescu R, editors. Physical modelling in geotechnics: proceedings of the international conference on physical modelling in geotechnics. Swets and Zeitlinger; 2002. p. 149–54. [3] Bond AJ. Behaviour of displacement piles in overconsolidated clays. PhD Thesis, Imperial College, London; 1989. [4] Bransby MF, Springman SM. 3D finite element modelling of piles groups adjacent to surcharge loads. Comput Geotech 1996;19:301–24. [5] Briaud JL, Tucker AM. Piles in sand: a method including residual stresses. J Geotech Eng 1984;110(11):1666–80. [6] Bustamante M. The pile loading test. In: Pilot G, editor. Foundation engineering, vol. 1. Presse de l’école Nationale des Ponts et Chaussées; 1982. p. 263–73. [7] Bustamante M, Gianeselli L. Predicting the bearing capacity of sheet piles under vertical load. In: Proceedings of the 4th international conference on piling and deep found, Stresa, Italy, 7–12 April; 1991. [8] Chow FC. Investigations in the behaviour of displacement piles for offshore foundations, PhD thesis, Imperial College, London; 1997. [9] Combarieu F. Estimation rapide de l’angle de frottement d’un sol granulaire au pressiomètre. Bull des Laborat des Ponts et Chaussées 1996;203:121–3. [10] Comodromos EM, Anagnostopoulos CT, Georgiadis MK. Numerical assessment of axial group response based on load test. Comput Geotech 2003;30:505–15. [11] Cubrinovski M, Ishihara K. Modelling of sand behaviour based on state concept. Soils Found 1998;38(3):115–27. [12] De Gennaro V. Etude du comportement mécanique des interfaces solstructure. Application à l’analyse du comportement des pieux. PhD thesis, Ecole Nationale des Ponts et Chaussées, Paris, France; 1999. p. 427. [13] De Gennaro V, Frank F. Elasto-plastic analysis of the interface behaviour between granular media and structure. Comput Geotech 2002;29:547–72. [14] De Gennaro V, Frank, F. Insight into the simulation of calibration chamber tests. In: Mestat, editor. Proceedings of the European conference on numerical methods in geotechnical engineering (NUMGE 2002), Presses de l’ENPC/LCPC, Paris; 2002b. p. 169–77. [15] De Gennaro V, Frank R. Elastoplastic modelling of the behaviour of the granular soil–structure interface. In: Frémond M, Maceri F, editors. Novel approaches in civil engineering. Springer Verlag; 2003. p. 133–58. [16] Desai CS, Zaman MM, Lightner JG, Sirirwardane HJ. Thin-layer element for interfaces and joints. Int J Numer Anal Meth Geomech 1984;8:19–43. [17] Dietz MS, Lings ML. Post peak strength of interfaces in a stress–dilatancy framework. J Geotech Geoenviron Eng, ASCE 2006;132(11). [18] Dijkstra J, Broere W, Frits van Tol A. Experimental investigation into the stress and strain development around a displacement pile; 2006. www.geo.citg.tudelft.nl/broere/pdf/dijkstra_experimental_pi_dfi2006.pdf. [19] Frank R, Guenot A, Humbert P. Numerical analysis of contacts in geomechnics. In: Proceedings of the fourth international conference on numerical methods in geomechanics, Edmonton, May 31–June 4; 1982. p. 37–45. [20] Frank R. Recent developments in the prediction of pile behaviour from pressiometer results. Rapport interne du LCPC; 1985. [21] Georgiadis K, Potts DM, Zdravkovic L. The influence of partial soil saturation on pile behaviour. Géotechnique 2003;53(1):11–25. [22] Hohberg J-M, Schweiger H. On the penalty behaviour of thin-layer elements. In: Proceedings of the fourth international symposium on numerical models in geometry (NUMOG IV), Swansea, UK; 1992. p. 241–48. [23] Humbert P. CESAR-LCPC, un code général de calcul par éléments finis. Bull. liaison Labo. Ponts et Chaussées 1989;160:112–6. [24] Li XS, Dafalias YF. Dilatancy for cohesionless soils. Géoetchnique 2000;50(4):449–60. [25] Lings ML, Dietz MS. An improved direct shear apparatus for sand. Géotechnique 2004;54(4):245–56. [26] Jardine R, Chow FC, Overy R, Standing J. ICP design methods for driven piles in sands and clays, 112. Thomas Telford Ltd; 2005. [27] Kuwano R. Laboratory tests on Dunkirk sand. Imperial College Internal Report; 1996. [28] Mabsout ME, Reese LC, Tassoulas L. Study of pile driving by finite element method. J Geotech Eng 1995;121(7):535–43. [29] Maqtadir A, Desai CS. Three-dimensional analysis of a pile-group foundation. Int J Numer Anal Meth Geomech 1986;10:41–58. [30] Mayne PM, Kulhawy FH. K0-OCR relationships in soils. J Geotech Eng Div, ASCE 1982;108(GT6):851–72. [31] Mohamedzein YEA, Mohamed MG, El Sharief AM. Finite element analysis of short piles in expansive soils. Comput Geotech 1999;24:231–43. [32] Muller H. Baugrunduntersuchung mit dem Pressiometerverfahren nach Menard, die Bautechnik, Heft 9, Berlin 1970. [33] Pande GN, Sharma KG. On joint/interface elements and associated problems of numerical ill-conditioning. Int J Numer Anal Meth Geomech 1979;3: 293–300.

I. Said et al. / Computers and Geotechnics 36 (2009) 6–19 [34] Randolph MF. Science and empiricism in pile foundation design. Géotechnique 2003;53(10):847–75. [35] Robinsky EI, Morrison CF. Sand displacement and compaction around model friction piles. Can Geotech J 1964;1(2):81–93. [36] Roscoe KH, Schofield AN, Thurairajh A. Yielding of clays in states wetter than critical. Géotechnique 1963;13(3):211–40. [37] Sharma KG, Desai CS. Analysis and implementation of thin-layer element for interfaces and joints. J Eng Mech, ASCE 1992;118(12):2442–62.

19

[38] Wakai A, Gose S, Ugai K. 3D elasto-plastic finite element analyses of pile foundations subjected to lateral loading. Soils Found 1999;39(1): 97–111. [39] Wehnert M, Vermeer PA. Numerical analyses of load tests on bored piles. Num Models Geomech (NUMOG IX) 2004:505–11. [40] White DJ, Bolton MD. Observing friction fatigue on a jacked pile. In centrifuge and constitutive modelling, Rotterdam; 2002. p. 347–54.